To see why such a result should hold, one has to make a further
step and go from groupoid-valued to \(\infty \) -groupoid-valued
deformation functors, and to think the whole problem in terms of
\(\infty \) -stacks [1]}, [2]}, [3]}. Indeed,
due to the presence of negative degree components in \(\mathcal {E}nd^*(\mathcal {E}^\cdot )\) , the groupoids \(\operatorname{Def}_{\mathcal {F}|_U}\)  are no more
equivalent to the Deligne groupoids \(\operatorname{Del}_{\mathcal {E}nd^*(\mathcal {E}^\cdot )(U)}\) ;
yet from the \(\infty \) -groupoid point of view it is natural to
expect that the stack \(\operatorname{Def}_{\mathcal {F}}\)  is locally homotopy
equivalent to the \(\infty \) -stack \(\operatorname{MC}_{\bullet }(\mathcal {E}nd^*(\mathcal {E}^\cdot ))\) . Then one reasons as in the locally free sheaf case,
using the fact that the Kan complexes-valued functor \(\operatorname{MC}_\bullet \) 
commutes with homotopy limits of DGLAs whose cohomology is
concentrated in positive degree [4]}:
\( \operatorname{Def}_{\mathcal {F}} \simeq \mathop {\rm holim}_{U\in \Delta _\mathcal {U}} \operatorname{Def}_{\mathcal {F}|_U}\simeq \mathop {\rm holim}_{U\in \Delta _\mathcal {U}} {\operatorname{MC}_{\bullet }(\mathcal {E}nd^*(\mathcal {E}^\cdot )(U))}\simeq {\operatorname{MC}_\bullet }(\mathop {\rm holim}_{U\in \Delta _\mathcal {U}} \mathcal {E}nd^*(\mathcal {E}^\cdot )(U)).\) 
